Srbija Posted August 14, 2022 Share #1 Posted August 14, 2022 (edited) Zsigmondy'S Theorem MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz Language: English | Size: 10.90 GB | Duration: 8h 24m Advancing Algebra to understand Number theory What you'll learn Proof of Zsigmondy's Theorem Applications of Zsigmondy's Theorem Lifting the Exponent Lemma Cyclotomic polynomials Complex roots of unity Mobius Inversion Number theory Algebra Requirements Modular arithmetic with prime numbers Sound algebra skills Description The story line that guides us is proving a theorem of Zsigmondy in number theory and seeing how it can be used to solve maths olympiad problems that would otherwise be quite difficult. To achieve this goal we first understand what I consider to be the most central topic in high school algebra which is omitted in high schools: cyclotomic polynomials. This sounds specialised but this is at the heart of all the algebra learned at high school such as factorising a difference of 2 squares or cubes. The cyclotomic polynomials gives a factorisation of x^n-1. When n is 2, this is just the difference of 2 squares. If you let the x be x/y then you really get x^2-y^2 after some easy manipulation. (x^n means x to the power of n)These lessons will be a very valuable part of a serious high school maths student or olympian.One of the really interesting features of this course is that the instructor learns the proof of the Zsigmondy Theorem with the students and you get to see how to educate yourself without further need to be taught. Overview Section 1: Introduction Lecture 1 Introduction Section 2: Polynomials Lecture 2 PST polynomials part 1 Lecture 3 PST polynomials part 2 Lecture 4 Irreducibility Section 3: Diophantine equations Lecture 5 PST 3: Diophantine equations Lecture 6 PST 3.11 Cyclotomic Recognition Section 4: Mobius inversion Lecture 7 Mobius inversion revisited Section 5: Cyclotomic polynomials Lecture 8 Cyclotomic polynomials skipping Mobius Lecture 9 Cyclotomic polynomials including Mobius Lecture 10 Cyclotomic recognition revisited Lecture 11 Infinitely many primes 1 mod n Lecture 12 2002 IMO Shortlist N3 Lecture 13 2006 IMO Shortlist N5 Section 6: LTE Lecture 14 Summary summarised Lecture 15 Problem 1 Lecture 16 2018 IMO Q4 Section 7: Zsigmond Prerequisites Lecture 17 Prerequisites stated and some proved Section 8: Zsigmondy Theorem Proof Lecture 18 Relate to Cyclotomic polynomials Lecture 19 Consolidate and continue Lecture 20 psi_n=lambda_nP_n Lecture 21 lambda_nP_n primes Lecture 22 lambda_n=1 and other cases Lecture 23 When a-b=1. QED Section 9: Zsigmondy Applications Lecture 24 a^n+b^n and other applications Lecture 25 Problems Maths olympiad students,Serious maths students,Serious students seeking proper foundation in algebra in high school Homepage Hidden Content Give reaction to this post to see the hidden content. Hidden Content Give reaction to this post to see the hidden content. Edited December 26, 2022 by Bad Karma Dead links removed Link to comment
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